Standard solutions: the associated Legendre functions
Pνμ⁡(z), Pν-μ⁡(z),
Qνμ⁡(z), and Q-ν-1μ⁡(z).
Pν±μ⁡(z) and Qνμ⁡(z) exist for
all values of ν, μ, and z, except possibly z=±1 and ∞,
which are branch points (or poles) of the functions, in general. When z is
complex Pν±μ⁡(z), Qνμ⁡(z), and
Qνμ⁡(z) are defined by
(14.3.6)–(14.3.10)
with x replaced by z: the principal
branches are obtained by taking the principal values of all the multivalued
functions appearing in these representations when z∈(1,∞), and by
continuity elsewhere in the z-plane with a cut along the interval
(-∞,1]; compare §4.2(i). The principal branches of
Pν±μ⁡(z) and Qνμ⁡(z) are real
when ν, μ∈ℝ and z∈(1,∞).